Gabriel's Horn - Apparent Paradox

Apparent Paradox

When the properties of Gabriel's Horn were discovered, the fact that the rotation of an infinite curve about the x-axis generates an object of finite volume was considered paradoxical. However, the explanation is that the bounding curve, is simply a special case–just like the simple harmonic series (Σx−1)–for which the successive area 'segments' do not decrease rapidly enough to allow for convergence to a limit. For volume segments however, and in fact for any generally constructed higher degree curve (e.g. y = 1/x1.001), the same is not true and the rate of decrease in the associated series is sufficiently rapid for convergence to a (finite) limiting sum.

The apparent paradox formed part of a great dispute over the nature of infinity involving many of the key thinkers of the time including Thomas Hobbes, John Wallis and Galileo Galilei.

Read more about this topic:  Gabriel's Horn

Famous quotes containing the words apparent and/or paradox:

    The apparent ease of California life is an illusion, and those who believe the illusion will live here in only the most temporary way.
    Joan Didion (b. 1935)

    The conclusion suggested by these arguments might be called the paradox of theorizing. It asserts that if the terms and the general principles of a scientific theory serve their purpose, i. e., if they establish the definite connections among observable phenomena, then they can be dispensed with since any chain of laws and interpretive statements establishing such a connection should then be replaceable by a law which directly links observational antecedents to observational consequents.
    —C.G. (Carl Gustav)